The mouse moves around the welcome screen, but clicking on a name has no effect. And as I don't know shortcut keys on this screen, I'm stuck.

Last time he had it, I disconnected and reconnected the mouse, fully turned off the power, waited a while and booted into Safe Mode. Everything worked. Rebooted into normal mode. Everything still worked. And I expect it will continue to work for at least 3 months.

Very unscientific, I must admit. And it's very unlikely it will help with you. So the more scientific answer is the sequence:
- see if you can boot into Safe mode with command prompt and do a System Restore back to when it still worked
- do a repair install of Windows
- do a clean install of Windows or use your recovery CD
- fix the broken hardware